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3. What do we mean when we say two quantities are in proportion ? It means that if: one of them doubles, the other one also doubles. one of them trebles, the other one also trebles. one of them x4, the other one also x4. one of them halves, the other one also halves. one of them ÷4, the other one also ÷4. Can you give examples of directly proportional quantities from every day life?

4. © T Madas Directly proportional quantities: They increase or decrease at the same rate More formally: Two variables are directly proportional if the ratio between them remains constant.

6. Two variables v and t are directly proportional. When t = 8, v =18. Write a formula which links v and t, in the form v = … Proportional vt

7. © T Madas Proportionality Constant vt v=kt This will be the formula when we find the value of k Two variables v and t are directly proportional. When t = 8, v =18. Write a formula which links v and t, in the form v = …

8. © T Madas vt v=kt v=kt 18=k x 8x 8 8k8k= k= 8 = 9494 =2.25 v=t 9494 v= 9t49t4 v=2.25t So: or: Two variables v and t are directly proportional. When t = 8, v =18. Write a formula which links v and t, in the form v = …

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10. In a chemistry experiment, the reaction time t is directly proportional to the mass m of the compound present. When the mass is 3 grams the reaction time is 0.2 seconds. 1.Write a formula which links t and m, in the form t = … 2.What is the reaction time when the mass is 8 grams? Proportional tm

11. © T Madas In a chemistry experiment, the reaction time t is directly proportional to the mass m of the compound present. When the mass is 3 grams the reaction time is 0.2 seconds. 1. Write a formula which links t and m, in the form t = … 2. What is the reaction time when the mass is 8 grams? tm Proportionality Constant t=km This will be the formula when we find the value of k

12. © T Madas In a chemistry experiment, the reaction time t is directly proportional to the mass m of the compound present. When the mass is 3 grams the reaction time is 0.2 seconds. 1. Write a formula which links t and m, in the form t = … 2. What is the reaction time when the mass is 8 grams? tm t=km t=km 0.2=k x 3x 3 3k3k= k= 3 = 2 30 ≈0.067 = 1 15 t=m 1 15 t= m 15 t≈0.067m So: or: t= m 15 using: t= 8 15 ≈ 0.53 s

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14. What do we mean when we say two quantities are inversely proportional ? It means that if: one of them doubles, the other one halves. one of them x3, the other one ÷3. one of them x4, the other one ÷4. one of them ÷2, the other one x2. one of them ÷10, the other one x10. Can you give an example of inversely proportional quantities from every day life?

15. © T Madas The Civic Centre is to be painted, so they call a firm of decorators. If this firm provide: 1 decorator 2 decorators 3 decorators 4 decorators 5 decorators 6 decorators 10 decorators 12 decorators 15 decorators 20 decorators 30 decorators 60 decorators 120 decorators will take 60 days for the job will take 30 days for the job will take 20 days for the job will take 15 days for the job will take 12 days for the job will take 10 days for the job will take 6 days for the job will take 5 days for the job will take 4 days for the job will take 3 days for the job will take 2 days for the job will take 1 day for the job will take ½ day for the job 1 x 60 2 x 30 3 x 20 4 x 15 5 x 12 6 x 10 10 x 6 12 x 5 15 x 4 20 x 3 30 x 2 60 x 1 120 x ½

16. © T Madas INVERSELY PROPORTIONAL QUANTITIES One increases at the same rate as the other one decreases. More formally: Two variables are inversely proportional if their product remains constant.

18. A variable P is inversely proportional to a variable A. When A = 2, P = 36. 1.Write a formula which links P and A, in the form P = … 2.Find the value of P when A is 2.5. Inversely Proportional P 1 A

19. © T Madas A variable P is inversely proportional to a variable A. When A = 2, P = 36. 1. Write a formula which links P and A, in the form P = … 2. Find the value of P when A is 2.5. P 1 A P 1 A =k x Proportionality Constant

20. © T Madas A variable P is inversely proportional to a variable A. When A = 2, P = 36. 1. Write a formula which links P and A, in the form P = … 2. Find the value of P when A is 2.5. P 1 A P 1 A =k x This will be the formula when we find the value of k P kAkA =

21. © T Madas A variable P is inversely proportional to a variable A. When A = 2, P = 36. 1. Write a formula which links P and A, in the form P = … 2. Find the value of P when A is 2.5. P 1 A P 1 A =k x P kAkA = P kAkA = 36 k 2 = k = 72 P A = So: P 72 A = using P 72 2.5 = 144 5 = 288 10 = = 28.8

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23. A variable F is inversely proportional to a variable t. When t = 3, F = 12. Find the value of t when F is 48. Inversely Proportional F 1 t

24. © T Madas A variable F is inversely proportional to a variable t. When t = 3, F = 12. Find the value of t when F is 48. F 1 t F 1 t =k x Proportionality Constant

25. © T Madas A variable F is inversely proportional to a variable t. When t = 3, F = 12. Find the value of t when F is 48. F 1 t F 1 t =k x This will be the formula when we find the value of k F ktkt =

26. © T Madas A variable F is inversely proportional to a variable t. When t = 3, F = 12. Find the value of t when F is 48. F 1 t F 1 t =k x F ktkt = F ktkt = 12 k 3 = k = 36 F t = So: F 36 t = using 48 36 t = 48 = t 36 = t 48 = 3 4

27. © T Madas A variable F is inversely proportional to a variable t. When t = 3, F = 12. Find the value of t when F is 48. Fxt=constant 12x3=36 ÷48=0.75 Since we do not require a formula in this example we could also have worked as follows: The product of inversely proportional quantities remains constant 48xt=36

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29. Sometimes we may be asked to set and solve problems involving direct or inverse proportion to the: square of a variable cube of a variable square root of a variable or simply combine 3 variables with direct and inverse proportion in the same problem.

31. A variable A is directly proportional to the square of another variable r. When r = 3, A = 36. Find the value of A, when r = 2.5 Ar A=kr 2r 2 36=k x 32x 32 9k9k= k= 4 A=4r 24r 2 So: using: 2 A=kr 2r 2 A=4r 24r 2 A=4 x 2.5 2 A=4 x 5252 2 A=4 x 25 4 A=

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33. A variable y is directly proportional to the SQUARE ROOT of another variable x. When x = 25, y = 3. Find the value of x, when y = 1.2 yx y=kx 3=k x 25 5k5k=3 k= 3535 y= 3535 So: using: y=kx = 0.6 x y= 3535 x 1.2= 3535 x 6 5 = 3535 x 5 x x 5 6=3 x =2 x =4 x

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35. A variable W is directly proportional to a variable m and inversely proportional to another variable t. When m = 2 and t = 8, W = 15. Find the value of W when m = 6 and t = 4. W 1 t W m t =k x So: m x W m t W km km t = W kmt kmt = 15 k x 2 8 = 15 2k 2k 8 = 2k2k = 120 k = 60 W 60m t = using: W 60m t = W 60 x 6 4 = W 360 4 = W = 90

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37. A variable F is directly proportional to a variable m and inversely proportional to the square of another variable r. When m = 10 and r = 2, F = 15. Find the value of F when m = 24 and r = 3. F 1 r 2r 2 F m r 2 =k x So: m x F m r 2r 2 F km km r 2r 2 = F kmr 2 kmr 2 = 15 k x 102 = 15 10k 4 = = 60 k = 6 F 6mr 2 6mr 2 = using: F 6mr 2 6mr 2 = F 6 x 24 3232 = F 144 9 = F = 16

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39. What does the graph of two directly proportional quantities looks like? Cost of packets of pens 3 pens cost £2

40. © T Madas Cost of packets of pens 12108642Cost (£) 181512963Number of pens 3 pens cost £2 Let us plot the information of this table in a graph

41. © T Madas Cost of packets of pens 12108642Cost (£) 181512963Number of pens 3 pens cost £2 4 8 12 16 20 24 12 10 8 6 4 2 0 pens £

42. © T Madas 4 8 12 16 20 24 12 10 8 6 4 2 0 pens £ when graphed the points of Directly Proportional Quantities: 1.always form a straight line through the origin 2.always form the corners of similar rectangles whose opposite corner is at the origin. 3.the line is a diagonal of every rectangle

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44. 12 8.16 10 6.8 8 5.44 5 3.4 v u 25211916 1714.2812.9210.88 The data above has been obtained from a chemistry experiment and concerns two quantities, u and v. Are u and v directly proportional quantities?

45. © T Madas 0 5 10 15 20 25 12 8.16 10 6.8 8 5.44 5 3.4 v u 25211916 1714.2812.9210.88 20 15 10 5 v the quantities u and v are directly proportional u

46. © T Madas u v v u 0 5 10 15 20 25 12 8.16 10 6.8 8 5.44 5 3.4 v u 25211916 1714.2812.9210.88 20 15 10 5 17 25 What is the gradient of the line? gradient= diff in y diff in x = 25 17 ≈ 1.47 the ratio between directly proportional quantities remains constant. Work the ratio v : u from the table and compare it with the gradient of this line. What would have happened if we plotted the data with the axes the other way round?

47. © T Madas u v 0 5 10 15 20 25 12 8.16 10 6.8 8 5.44 5 3.4 v u 25211916 1714.2812.9210.88 20 15 10 5 25 What is the gradient of the line? gradient= diff in y diff in x = 25 17 ≈ 1.47 the ratio between directly proportional quantities remains constant. Work the ratio v : u from the table and compare it with the gradient of this line. What would have happened if we plotted the data with the axes the other way round? 17 25 = 0.68 u : v

48. © T Madas 12 8.16 10 6.8 8 5.44 5 3.4 v u 25211916 1714.2812.9210.88 uv u=kv u=kv 3.4=k x 5x 5 5k5k= k= 5 = 6.8 10 =0.68 u=0.68v So: We could obtain a formula linking u and v The proportionality constant is the gradient of the line in the graph v= 1 0.68 or u v≈1.47 u

49. © T Madas 12 8.16 10 6.8 8 5.44 5 3.4 v u 25211916 1714.2812.9210.88 v u 0 5 10 15 20 25 20 15 10 5 v≈1.47 u

50. © T Madas 12 8.16 10 6.8 8 5.44 5 3.4 v u 25211916 1714.2812.9210.88 u v 0 5 10 15 20 25 20 15 10 5 u=0.68 v

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52. What does the graph of two inversely proportional quantities looks like? 1 decorator takes 24 days to finish a job

53. © T Madas 1 decorator takes 24 days to finish a job 1234681224Days 2412864321No of decorators 0 5 10 15 20 25 20 15 10 5 decorators days

54. © T Madas 0 5 10 15 20 25 20 15 10 5 decorators days The graphed points of Inversely Proportional Quantities: 1.always lie on a curve like the one shown below. 2.always form the corners of rectangles of constant area whose opposite corner is at the origin. Hyperbola

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56. The data above has been obtained from the physics department and concerns two quantities, P and A. Are P and A inversely proportional quantities? 3.754.567.51011.251518 A 242015129865 P

57. © T Madas 0 5 10 15 20 25 3.754.567.5911.251518 A 2420151210865 P 20 15 10 5 P A Hyperbola

58. © T Madas 1234567 4.5 4 3.5 3 2.5 2 1.5 1 0.5 When plotted, Inversely Proportional quantities, always show as Hyperbolas.

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60. Suppose we have a formula which contains 2 or more variables. The data which produced this formula is not available. Is it possible to establish if variables are directly proportional or inversely proportional? This is how this is done.

61. © T Madas Directly proportional to variables which appear in the numerator of the R.H.S Inversely proportional to variables which appear in the denominator of the R.H.S The variable for which the formula is solved for is: s v t = vsv and s are directly proportional v 1t1t v and t are inversely proportional c s=vtsts and t are directly proportional

62. © T Madas Directly proportional to variables which appear in the numerator of the R.H.S Inversely proportional to variables which appear in the denominator of the R.H.S The variable for which the formula is solved for is: V m gh = VmV and m are directly proportional m 1g1g m and g are inversely proportional c V=mgh V h mg = c VgV and g are directly proportional VhV and h are directly proportional m 1h1h m and h are inversely proportional h 1g1g h and g are inversely proportional

63. © T Madas Directly proportional to variables which appear in the numerator of the R.H.S Inversely proportional to variables which appear in the denominator of the R.H.S The variable for which the formula is solved for is: GMm F r 2r 2 = FGF and G are directly proportional FMF and M are directly proportional FmF and m are directly proportional F 1r 21r 2 F is inversely proportional to the square of r To get relationships between any other 2 variables we appropriately rearrange the formula.

64. © T Madas Directly proportional to variables which appear in the numerator of the R.H.S Inversely proportional to variables which appear in the denominator of the R.H.S The variable for which the formula is solved for is: 4 V 3 = Vr 3r 3 V is directly proportional to the cube of r Rearranging the formula for r gives: π r 3r 3 V π Because π is not a v _______ ; π is a c_______ n______ ariable onstant umber 3 3V3V 4 r π = CHALLENGE r is directly proportional to the c___ r___ of __ ube oot V

65. © T Madas Directly proportional to variables which appear in the numerator of the R.H.S Inversely proportional to variables which appear in the denominator of the R.H.S The variable for which the formula is solved for is: u + v S t = S 1t1t S and t are inversely proportional Su Because u and v are not in a product Sv S is directly proportional to the sum of u and v Su + v

66. © T Madas Now a harder, worded proportionality problem

67. © T Madas 60 workers, working a 9 hour day produce 720 toys a day. 1.Find a formula which relates the number of workers w, the number of hours they work h and the number of toys T they produce. 2.How many hours a day, do 90 workers need to work if they are to produce 1020 toys? The formula must contain the 3 variables w, h and T Suppose that: the workers work a constant number of hours per day Then: If we double the workers, ___________________________ the toys produced will also double Toys and workers are directly proportional quantities Tw Suppose that: we keep the number of workers constant Then: doubling the hours they work, ___________________________ the toys produced will also double Toys and hours are directly proportional quantities Th

68. © T Madas 60 workers, working a 9 hour day produce 720 toys a day. 1. Find a formula which relates the number of workers w, the number of hours they work h and the number of toys T they produce. 2. How many hours a day, do 90 workers need to work if they are to produce 1020 toys? The formula must contain the 3 variables w, h and T TwThTh

69. © T Madas 60 workers, working a 9 hour day produce 720 toys a day. 1. Find a formula which relates the number of workers w, the number of hours they work h and the number of toys T they produce. 2. How many hours a day, do 90 workers need to work if they are to produce 1020 toys? The formula must contain the 3 variables w, h and T TwTh Tw h Tk w = h T=kwh 720=k x 60 540k=720 k= 540 = 4343 x 9x 9 T w = h 4343 Check that it works

70. © T Madas 60 workers, working a 9 hour day produce 720 toys a day. 1. Find a formula which relates the number of workers w, the number of hours they work h and the number of toys T they produce. 2. How many hours a day, do 90 workers need to work if they are to produce 1020 toys? The formula must contain the 3 variables w, h and T TwTh Tw h Tk w = h T w = h 4343 T w = h 4343 1020 x 90 = x h 4343 1020 120 = h h 1020 120 = h 8.5 hours =

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72. © T Madas Three variables u, v and w are related by a formula. The following table gives some of the values that these three variables can take: Obtain the formula linking these variables, solved for u. 54321111w 121520308642v 120 161284u uv

73. © T Madas Three variables u, v and w are related by a formula. The following table gives some of the values that these three variables can take: Obtain the formula linking these variables, solved for u. 54321111w 121520308642v 120 161284u v 1 w v 1 w v u w =k x So: u x v u w v ku ku w = v kuw kuw = 2 k x 4 1 = 4k4k = 2 k = v u w = uv 1212 u v = w u 2 = v w

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